Digital sums and functional equations (Q2882735)

Let \(s(j)\) be the sum of digits functions of the binary expansion of \(j\). The author presents a new method to obtain an explicit representation of NEWLINE\[NEWLINES(n)=\sum_{j=0}^{n-1}s(j),NEWLINE\]NEWLINE which was already proven by \textit{J. R. Trollope} [Math. Mag. 41, 21--25 (1968; Zbl 0162.06303)] and \textit{H. Delange} [Enseign. Math. (2) 21, 31--47 (1975; Zbl 0306.10005)].NEWLINENEWLINEThis is done by stating simple recursive formulas for \(S(n+2^{[\log_{2}n]})\) and similar terms (where \([x]\) is the largest integer less than or equal to \(x\)). These are then combined to a system of two functional equations, for which only one continuous solution exists.NEWLINENEWLINEThe resulting explicit representation is NEWLINE\[NEWLINE\frac{1}{n}S(n)=\frac{\log_{2}n}{2}-\frac{\log_{2}(x+1)}{2}+\frac{1}{x+1}F(x)NEWLINE\]NEWLINE where \(F(x)\) is a continuous function on the interval \([0,1]\) and NEWLINE\[NEWLINEx=\frac{n-2^{[\log_{2}n]}}{2^{[\log_{2}n]}}.NEWLINE\]NEWLINENEWLINENEWLINEThis method is also used to obtain explicit representations for the moment generating function of the sum of digits function NEWLINE\[NEWLINE\sum_{j=0}^{n-1}\exp(t\cdot s(j)),NEWLINE\]NEWLINE its \(k\)-th moment NEWLINE\[NEWLINE\sum_{j=0}^{n-1}s(j)^{k},NEWLINE\]NEWLINE and the same for the number of zeros of the binary expansion.